How is Statistics Different

from Mathematics, and Why Should Teachers

Care?Allan Rossman and Beth Chance

Cal Poly - SLO

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Context matters

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weight (lbs)

Weights of members of 2000 U.S. Men’s Olympic Rowing team

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125

115

105

95

85

75

65

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Context matters

Gesell (aptitude) score vs. age (in months) of first speaking

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age of first speakingG

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Context matters

Without outliers

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age of first speaking

Ges

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Context matters

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level

proportion

patientspamphlets

Are the cancer pamphlets written at appropriate levels

to be read and understood by the cancer patients?

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Measurement matters

Unemployment Intelligence Highway safety Authoritarian personality Memory ability Ambidextrous-ness Teaching effectiveness Pace of life

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Measurement matters

Is a geographic region’s “pace of life” associated with its heart disease rate? Average walking speed of pedestrians over a distance of

60 feet during business hours on a clear summer day along a main downtown street

Average time a sample of bank clerks take to make change for two $20 bills or to give $20 bills for change

Average ratio of total syllables to time of response when asking a sample of postal clerks to explain the difference between regular, certified, and insured mail

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Measurement matters

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Data collection design matters “Ladies, do you give more emotional support

to your husband or boyfriend than you receive in return?” Study A: 96% of a sample of 4500 said “yes” Study B: 44% of a sample of 767 said “yes” Which study do you have more confidence in?

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Data collection design matters “Ladies, do you give more emotional support

to your husband or boyfriend than you receive in return?” Study A: Sociologist Shere Hite distributed over

100,000 questionnaires through women’s groups, got 4500 responses

Study B: ABC News - Washington Post conducted interviews with a random sample of 767 women

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Data collection design matters Study A:

Group 1: 42 successes in 61 trials (.689) Group 2: 30 successes in 62 trials (.484) P-value = .011

Study B: Group 1: 806 successes in 908 trials (.888) Group 2: 614 successes in 667 trials (.920) P-value = .015

Study C: Group 1: 3274 successes in 3775 trials (.867) Group 2: 6438 successes in 7225 trials (.891) P-value = .000

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Data collection design matters Study A: Social experiment that randomly

assigned three- and four-year-old children to 2 years of preschool instruction or control group Strong evidence of causal benefit of preschool

Study B: Observational study of court records, comparing violent crime rates among those abused as children and control group Strong evidence of association, but no causal link

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Data collection design matters Study C: On-time flight arrivals in one month

for Alaska Airlines and America West America West had higher on-time arrival rate Airport-by-airport analysis reveals that Alaska had

higher on-time arrival rate for every airport America West had most flights to Phoenix, with very

high on-time arrival rate Alaska had most flights to Seattle and SF, with lower on-

time arrival rates Lurking, confounding variable

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Data analysis requires substantial judgment Outliers

Should outlier(s) be removed? Should I apply a more resistant method?

Technical conditions Are they satisfied?

Never perfectly, but close enough? Is the technique robust enough to proceed anyway?

Transformations Should I apply one at all? How do I choose which one to use?

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Inductive vs. deductive reasoning Mathematics

Deductive reasoning Logical thinking Problem solving Probability

Statistics Inductive reasoning, conditional reasoning Draw conclusions from data Make inferences from data

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Uncertainty abounds!

“Statistics is never having to say you’re certain.”

“You never know. You really never know. Really.”

Correct “We have strong evidence that ….” “The data strongly suggest that …”

Incorrect “The data prove that …”

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Uncertainty abounds!

Rarely is there a definitive conclusion Often there is not even a definitive approach

to a problem

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Terminology crucial?

Also true in mathematics, but … Everyday language has technical meaning

Bias, sample, statistic, accuracy, precision, confound, correlation, confident, significant, normal, random

Analogous to studying foreign language

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Communication crucial

Explanations in layperson terms essential Statistics is a consulting enterprise Must constantly interact with clients whose

technical skills vary greatly Must elicit from them what problem is Must communicate to them results and conclusions

Most AP Students will be consumers not producers of statistics

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Much newer discipline

Think about when these ideas/tools were first developed Geometry, logic, proof, trigonometry, function, calculus Boxplot, stemplot, randomized comparative experiment,

least squares regression, t-test Much mathematics that we teach is millenia old

All is at least many centuries old Some statistics that we teach is 100 years old

Much is a few decades old

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Summary: How is Statistics Different from Mathematics? Context matters

Question of interest matters Measurement method matters Data collection design matters

Substantial judgment involved Outliers, resistance Technical conditions, robustness Transformations

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How is Statistics Different from Mathematics? (Summary) Inductive vs. deductive reasoning Uncertainty abounds

Few definitive conclusions Few definitive approaches

Terminology crucial Everyday phrases adopt technical meanings

Communication crucial Explanation in layperson terms essential

Much newer discipline

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Why should teachers care?

Different preparation needed Real data, meaningful contexts Technology Understand different kinds of concepts, skills

Often weren’t taught in teacher preparation Development of students’ communication skills Successful teaching strategies in other classes

may not work as well here

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Why should teachers care?

Different for students Research shows difficulty of reasoning with

uncertainty Many students very uncomfortable with

uncertainty, lack of definitive conclusions, need for detailed explanations, individual interpretation

Challenge of promoting healthy skepticism without extremes of cynicism or naïve acceptance

Many mathematically strong students will be frustrated But many less stellar math students will be empowered

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How is Statistics Different from Mathematics? (Final Analysis) It’s more fun!!